Question 3 :
State whether the given statement is True or False :<br/>$2\sqrt { 3 }-1 $ is an irrational number.

Question 6 :
State True or False:$4\, - \,5\sqrt 2 $ is irrational if $\sqrt 2 $ is irrational.

Question 8 :
State whether the given statement is True or False :<br/>$\sqrt { 3 } +\sqrt { 4 }&#160;$ is an irrational number.

Question 11 :
In a division operation the divisor is $5$ times the quotient and twice the remainder. If the remainder is $15,$ then what is the dividend?

Question 12 :
When a natural number x is divided by 5, the remainder is 2. When a natural number y is divided by 5, the remainder is 4. The remainder is z when x+y is divided by 5. The value of $\dfrac { 2z-5 }{ 3 } $ is

Question 13 :
Solve the following equations:<br/>$x + \dfrac {4}{y} = 1$,<br/>$y + \dfrac {4}{x} = 25$.Then $(x,y)=$

Question 14 :
The&#160;&#160;linear equation, such that each point on its graph has an ordinate $3$ times its abscissa is $y=mx$. Then the value of&#160;$m$ is<br/>

Question 15 :
Let PS be the median of the triangle with vertices $P\left( 2,2 \right), Q\left( 6,-1 \right), R\left( 7,3 \right).$The equation of the line passing through $\left( 1,-1 \right)$and parallel to PS is

Question 17 :
The graph of the linear equation $2x -y = 4$ cuts x-axis at

Question 18 :
If $6$ kg of sugar and $5$ kg of tea together cost Rs. $209$ and $4$ kg of sugar and $3$ kg of tea together cost Rs. $131$, then the cost of $1$ kg sugar and $1$ kg tea are respectively<br/>

Question 20 :
To save on helium costs, a balloon is inflated with both helium and nitrogen gas. Between the two gases, the balloon can be inflated up to $8$ liters in volume. The density of helium is $0.20$ grams per liter, and the density of nitrogen is $1.30$ grams per liter. The balloon must be filled so that the volumetric average density of the balloon is lower than that of air, which has a density of $1.20$ grams per liter. Which of the following system of inequalities best describes how the balloon will be filled, if $x$ represents the number of liters of helium and $y$ represents the number of liters of nitrogen ?

Question 21 :
Solve: $\displaystyle \frac{3}{x}\, -\,&#160;\displaystyle \frac{2}{y}\, =\, 0$ and $\displaystyle \frac{2}{x}\, +\,&#160;\displaystyle \frac{5}{y}\, =\, 19$<br/>Hence, find 'a' if $y\, =\, ax\, +\, 3$

Question 22 :
What is the value of $a$ for the following equation: $3a + 4b = 13$ and $a + 3b = 1$?&#160;(Use cross multiplication method).<br/>

Question 23 :
Father's age is three times the sum of ages of his two children. After $5$ years his age will be twice the sum of ages of two children. Find the age of father.<br/>

Question 24 :
The equations of two equal sides of an isosceles triangle are $ 3x + 4y = 5 $and $4x - 3y = 15$. If the third side passes through $(1, 2)$, its equation is

Question 25 :
Based on equations reducible to linear equations, solve for $x$ and $y$:<br/>$\dfrac {x-y}{xy}=9; \dfrac {x+y}{xy}=5$<br/>

Question 26 :
Is the following equation a quadratic equation?$\displaystyle 3x + \frac{1}{x} - 8 = 0$

Question 27 :
The difference between the product of the roots and the sum of the roots of the quadratic equation $6x^{2} - 12x + 19 = 0$ is

Question 28 :
In a rectangle the breadth is one unit less than the length and the area is $12$ sq.units. Find the length of the rectangle.

Question 29 :
If&#160;$\displaystyle \frac{5x+6}{\left ( 2+x \right )\left ( 1-x \right )}=\frac{a}{2+x}+\frac{b}{1-x}$, then the values of a and b respectively are

Question 30 :
Obtain a quadratic equation whose roots are reciprocals of the roots of the equation $x^2-3x - 4 =0$.

Question 31 :
The nature of roots of the equation<br>$\left( a+b+c \right) { x }^{ 2 }-2\left( a+b \right) x+\left( a+b-c \right) =0\left( a,b,c\epsilon Q \right) $

Question 32 :
The values of k&#160;for which the roots are real and equal of the following equation<br/>$4x^2$ - 3kx + 1 = 0 are&#160;$k = \pm \dfrac{4}{3}$<br/>

Question 33 :
Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals? Why?<br><br>

Question 34 :
If $x_1$ and $x_2$ are the roots of $3x^2 - 2x - 6 = 0$, then $x_1^2 + x_2^2$ is equal to

Question 35 :
Determine the nature of roots of the given equation from its discriminant.<br/>$x^{2}\, +\, 3\sqrt2x\, -\, 8\, =\, 0$

Question 36 :
If&#160;$\displaystyle r_{1}\:$ and $ r_{2}$ are the roots of&#160;$\displaystyle x^{2}+bx+c=0$ and&#160;$\displaystyle S_{0}=r_{1}^{0}+r_{2}^{0}$,&#160;$\displaystyle S_{1}=r_{1}+r_{2}$ and&#160;$\displaystyle S_{2}=r_{1}^{2}+r_{2}^{2}$, then the value of&#160;$\displaystyle S_{2}+bS_{1}+cS_{0}$ is

Question 37 :
Divide 20 into 2 parts such that the product of 2 numbers is 36.